Question

Which of the following statement(s) is(are) correct?

• A. If f is a one-one mapping from set A to A, then f is onto.
• B. If f is an onto mapping from set A to A, then f is one-one.
• C. Let f and g be two functions defined from $R\to R$ such that gof is injective, then f must be injective.
• D. If set A contains 3 elements while set B contains 2 elements, then total number of functions from A to B is 8

Answer 1

Answer: (C), (D)

(A) Let $f:N\to N$ such that $f(x)=2x$. Clearly $f$ is one-one but not onto.
Note: If $f$ is a one-one mapping from set $A$ to $A$, then $f$ is onto only if $A$ is finite set.
(B) $f:R\to R$ such that $f(x)=x^3-x^2-4x+4$. Clearly $f(-2)=f(2)=f(1)=0$.
Hence $f$ is many one but since it is an odd degree polynomial therefore its range is $R$ hence it is onto.
Note: If $f$ is a onto mapping from set $A$ to $A$ then $f$ is one-one only if $A$ is finite set.
(C) Suppose $f$ is not one-one then there are atleast two real numbers $x_1,x_2\in R,x_1\ne x_2$ such that $f\left(x_1\right)=f\left(x_2\right)$
$\therefore g\left(f\left(x_1\right)\right)=g\left(f\left(x_2\right)\right)$
i.e. gof is not one-one which is a contradiction to the given hypothesis that gof is one-one.
Hence $f$ must be one-one.
(D) Clearly, total number of functions from A to $B=2\times 2\times 2=8$.